If the button is still not visible on the People page, check if the 'Out of office' button is added to the User Page layout:
1. Go to Setup > Object Manager > User > User Page Layouts
2. From there, check if the 'Out of Office' button is added under 'Salesforce Mobile and Lightning Experience Actions'.
Once the Out of Office is set for a user, when selecting the user in a Chatter post with the "@name" specification, the Out of Office notice will appear along with the dates the user selected.
Salesforce Help - Set an Out of Office Message in Chatter
000381301

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